Introduction to Linear Transformations

A linear transformation T (on a vector $ \overrightarrow{v} $) is a transformation that can be represented as a matrix A times $ \overrightarrow{v} $.

The linear transformation can be thought of as using the matrix A to output some other vector when $ \overrightarrow{v} $ is put in, by simply using matrix multiplication to produce the output, $ A\overrightarrow{v} $.

One can think of T as a linear transformation, because it satisfies the following two equalities:

  • $ T(\overrightarrow{v}+\overrightarrow{w})=T(\overrightarrow{v})+T(\overrightarrow{w}) $ for vectors v and w.
  • $ T(k\overrightarrow{v})=k*T(\overrightarrow{v}) $ for any vector v and scalar k.

In other words, you can add vectors, then take the transformation, or take the transformations separately and add them together at the end. Similarly, you can multiply by a constant and then take the transformation, or take the transformation, and then multiply by a constant.

In fact, any transformation that satisfies these two properties is a linear transformation.

Identifying linear transformations with matrices

Because each linear transformation has an associated matrix, when you talk about properties of a linear transformation, you can think about similar properties of the associated matrix. Technically a matrix and a linear transformation have different meanings, but since the matrix is what defines the linear transformation, they can be thought of in a similar way.

For example, the function T that takes (x,y) to (x+2y, 3x+4y) is a linear transformation. It can be associated with the matrix A:

$ \begin{bmatrix} 1 & 2\\ 3 & 4\end{bmatrix} $

Thus, T(v)=Av. One can check this using matrix multiplication. T(v)=Av=A*(x,y)=(1*x+2*y, 3*x+4*y).

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